Hopf bifurcation of hybrid van der Pol oscillators
type: Item
instance of: scholarly article
title: Hopf bifurcation of hybrid van der Pol oscillators
zbMATH Open document ID: 1386.34070
author: Leonardo Herrera
author: Óscar Montaño
author: Yury V. Orlov
published in: Nonlinear Analysis. Hybrid Systems
publication date: 2017-11-16T00:00:00Z
review text: The authors study the dynamics of the modified Van der Pol equation \[ \ddot{x} + \varepsilon \biggl(\biggl(x^2 + {\dot{x}^2 \over \mu^2}\biggr)-\rho^2\biggr) \dot{x} + \mu^2 x = 0 \] subject to the constraint \(x \geq 0\). When \(x(t)\) hits the boundary, an impact with restitution coefficient \(e\) occurs: \[ \dot{x}(t^+) = -e \dot{x}(t^-). \] The authors prove that for certain values of the parameter \(\alpha=\varepsilon\rho^2/\mu\) the origin loses stability and a periodic solution bifurcates at this Neimark-Sacker bifurcation. Using some numerical software, the authors calculate the Neimark-Sacker bifurcation boundary in the \((e, \alpha)\)-plane and provide a (quite inaccurate) interpolation formula. It should be noted, that this stability boundary could be given explicitely by \[ \alpha_c = {-2\log e \over \sqrt{\pi^2 + \log^2 e}}. \] Finally, the authors demonstrate that by varying the parameters different solution amplitudes can be enforced.
reviewed by: Alois Steindl
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 34C05
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 6807803
zbMATH Keywords: Van der Pol oscillator
zbMATH Keywords: unilateral constraints
zbMATH Keywords: Neimark-Sacker bifurcation
zbMATH Keywords: Hopf bifurcation
zbMATH Keywords: periodic solutions
MaRDI profile type: Publication
full work available at URL: j.nahs.2017.05.003
OpenAlex ID: W2735887459
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DOI: 10.1016/J.NAHS.2017.05.003
description: scientific article; zbMATH DE number 6807803